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A number consisting entirely of ones is called a repunit. We shall define \(R(k)\) to be a repunit of length \(k\); for example, \(R(6)=111111\). Given that \(n\) is a positive integer and \({\rm GCD}(n, \;10)=1\), it can be shown that there always exists a value, \(k\), for which \(R(k)\) is divisible by \(n\), and let \(A(n)\) be the least such value of \(k\); for example, \(A(7)=6\) and \(A(4..

A number consisting entirely of ones is called a repunit. We shall define \(R(k)\) to be a repunit of length \(k\); for example \(R(6)=111111\). Given that \(n\) is positive integer and \({\rm GCD}(n, \;10) =1\), it cane be shown that there always exists a value, \(k\), for which \(R(k)\) is divisible by \(n\), and let \(A(n)\) be the least such value of \(k\); for example, \(A(7)=6\) and \(A(41..

A hexagonal tile with number \(1\) is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering the tiles \(2\) to \(7\) in an anti-clockwise direction. New rings are added in the same fashion, with the next ring being numbered \(8\) to \(19\), \(20\) to \(37\), \(38\) to \(61\), and so on. The diagram below shows the first three rings. By finding the difference between..